Today, I have made an observation which sounds very primitive when you state it:

in playing piano you must be totally concentrated on the music.
What would you mean by total concentration. Let me give an example: when you drive a car, you can listen to music, you could talk to passengers (which is not allowed for bus drivers). Anyway, if you lose the concentration on the road ahead of you for even a second, the car will still drive in straight line and if there is no car in front of you suddenly stopping, nothing bad will happen.
Many people are talking about multi-tasking. But that functions only on a time slicing method, you are alternately thinking about the various tasks.
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I was always admiring how players could play piano and talk at the same time. My father could do that. Famous entertainers can do that. I can manage to say one sentence and then recover to concentration. I believe that my capability of playing prima vista and playing difficult pieces reading the scores is basically just the technique of concentration just on the music.
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For tomorrow's performance I have "learned" a small piece by heart. I can even lay in bed in play it to myself in my head. However, when I sit at the piano playing and just one strange thought crosses my head, I am doomed for an error. This is not because of technical difficulties. Obviously, my mind must be connected to the fingers in a very direct mode. (On a computer it would be called DMA, direct memory access.) Any irritation on that mode would cause an interruption.
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Of course, after more time of practicing you might achieve a sort of mechanical routine. With some exercises it is even intended and probably achieved.
For me that is not so easy. And I would even say, that once I have achieved that mechanical perfection, the musical attraction for this piece fades away.
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Afterthough: the total concentration was possible for me when I worked on a program, actually direct coding. I could keep the concentration for two or three hours. I believe that the state of total concentration is related to "the flow" which is described by Mihály Csíkszentmihályi.